We study a system made from <math altimg='si6.svg'><mrow><msup><mi>K</mi><mrow><mo>*</mo><mo>+</mo></mrow></msup><msup><mi>D</mi><mrow><mo>*</mo><mo>+</mo></mrow></msup><msup><mi>K</mi><mrow><mo>*</mo><mo>+</mo></mrow></msup></mrow></math> with charge 3, isospin <math altimg='si7.svg'><mrow><mi>I</mi><mo linebreak='goodbreak'>=</mo><mn>3</mn><mo linebreak='goodbreak'>/</mo><mn>2</mn></mrow></math>, spin <math altimg='si8.svg'><mrow><mi>J</mi><mo linebreak='goodbreak'>=</mo><mn>3</mn></mrow></math>, and a quark content of <math altimg='si9.svg'><mrow><mi>c</mi><mover accent='true'><mi>d</mi><mo>&#175;</mo></mover><mover accent='true'><mi>s</mi><mo>&#175;</mo></mover><mi>u</mi><mover accent='true'><mi>s</mi><mo>&#175;</mo></mover><mi>u</mi></mrow></math>, which make it highly exotic relative to the standard <math altimg='si10.svg'><mrow><mi>q</mi><mover accent='true'><mi>q</mi><mo>&#175;</mo></mover></mrow></math> structure of mesons. The interaction of the three body system is obtained starting from a cluster of <math altimg='si11.svg'><mrow><msup><mi>D</mi><mrow><mo>*</mo><mo>+</mo></mrow></msup><msup><mi>K</mi><mrow><mo>*</mo><mo>+</mo></mrow></msup></mrow></math> in <math altimg='si12.svg'><mrow><mi>I</mi><mo linebreak='goodbreak'>=</mo><mn>1</mn></mrow></math> and <math altimg='si13.svg'><mrow><mi>J</mi><mo linebreak='goodbreak'>=</mo><mn>2</mn></mrow></math>, that in different works has been found bound, and adding to it an extra <math altimg='si14.svg'><msup><mi>K</mi><mrow><mo>*</mo><mo>+</mo></mrow></msup></math> with spin aligned with those of the vectors of the cluster. We find that the <italic>K</italic>* <italic>K</italic>* interaction in <math altimg='si12.svg'><mrow><mi>I</mi><mo linebreak='goodbreak'>=</mo><mn>1</mn></mrow></math> and <math altimg='si13.svg'><mrow><mi>J</mi><mo linebreak='goodbreak'>=</mo><mn>2</mn></mrow></math> is repulsive, but its strength is small compared to that of <math altimg='si11.svg'><mrow><msup><mi>D</mi><mrow><mo>*</mo><mo>+</mo></mrow></msup><msup><mi>K</mi><mrow><mo>*</mo><mo>+</mo></mrow></msup></mrow></math> in <math altimg='si12.svg'><mrow><mi>I</mi><mo linebreak='goodbreak'>=</mo><mn>1</mn></mrow></math> and <math altimg='si13.svg'><mrow><mi>J</mi><mo linebreak='goodbreak'>=</mo><mn>2</mn></mrow></math>, such that we find a three body state bound by about 100&#8201;MeV with respect to the mass of a <math altimg='si14.svg'><msup><mi>K</mi><mrow><mo>*</mo><mo>+</mo></mrow></msup></math> and the <math altimg='si11.svg'><mrow><msup><mi>D</mi><mrow><mo>*</mo><mo>+</mo></mrow></msup><msup><mi>K</mi><mrow><mo>*</mo><mo>+</mo></mrow></msup></mrow></math> cluster. The width of the state, of about 10&#8201;MeV, is much smaller than the binding, which facilitates its observation. We suggest to find that state by measuring the invariant mass of <italic>KDK</italic>*, something feasible in present experimental facilities.

China

Wen-Hao Jia

Pei-Shen Su

Wei-Hong Liang

Spain

Raquel Molina

Eulogio Oset

Superexotic <math altimg='si6.svg'><mrow><msup><mi>K</mi><mrow><mo>*</mo><mo>+</mo></mrow></msup><msup><mi>D</mi><mrow><mo>*</mo><mo>+</mo></mrow></msup><msup><mi>K</mi><mrow><mo>*</mo><mo>+</mo></mrow></msup></mrow></math> bound state

We study the decay $$\Lambda _c^+ \rightarrow \Lambda \pi ^+ \pi ^+ \pi ^-$$ , focusing on the production of the $$\Sigma ^*(1430)$$ resonance observed by the Belle Collaboration. Interpreted as a dynamically generated state from meson-baryon interactions in the chiral unitary approach, the $$\Sigma ^*(1430)$$ signal is shown to be enhanced by a triangle singularity involving intermediate $$K^{*-}$$ , $\textit{p}$, and $${\bar{K}}^0$$ states. This mechanism leads to a sharp peak near 1434 MeV in the $$\pi ^+ \Lambda $$ invariant mass distribution, in agreement with the experimental observations, and predicts a secondary peak around 1875 MeV in the $$\pi ^- \Sigma ^*(1430)$$ spectrum tied to the triangle singularity. We also estimate the branching ratio of $$\Lambda _c^+ \rightarrow \pi ^+ \pi ^- \Sigma ^*(1430)$$ to be about $$3.5 \times 10^{-4}$$ . The results for the branching ratio and the $$\pi ^- \Sigma ^*(1430)$$ mass distributions are predictions of the theoretical approach, which could be tested with reanalysis of existing data.

Yi-Yao Li

Jing Song

The $$\Lambda _c^+\rightarrow \Lambda \pi ^+\pi ^+\pi ^-$$ reaction, and a triangle singularity producing the $$\Sigma ^*(1430)$$ state

<p>We have studied the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>N</mi><msup><mrow><mi>D</mi></mrow><mrow><mo>*</mo></mrow></msup><msup><mrow><mover accent="true"><mrow><mi>K</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mo>*</mo></mrow></msup></mrow></math> system in the framework of the fixed center approximation to the Faddeev equations, taking the exotic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>D</mi><mo>*</mo></msup><msup><mover accent="true"><mi>K</mi><mo stretchy="false">¯</mo></mover><mo>*</mo></msup></math> system as the cluster and allowing the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi></math> to interact with the components of the cluster. Previous studies have determined the existence of three states of spin 0, 1, 2 for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>D</mi><mo>*</mo></msup><msup><mover accent="true"><mi>K</mi><mo stretchy="false">¯</mo></mover><mo>*</mo></msup></math> system, the one of spin 0 associated to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>X</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>2900</mn><mo stretchy="false">)</mo></math> state observed by the LHCb Collaboration. From this perspective, we find five states with total spin <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>1</mn><mo>/</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>/</mo><mn>2</mn><mo>,</mo><mn>5</mn><mo>/</mo><mn>2</mn></math>, with bindings from 10 to 30 MeV and widths below 60 MeV, which could be well identified. We also discuss the decay channels of these states that should help in future experimental searches for these states.</p>

Qing-Yu Zhai

Li-Sheng Geng

Study of the exotic three-body <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>N</mi><msup><mrow><mi>D</mi></mrow><mrow><mo>*</mo></mrow></msup><msup><mrow><mover accent="true"><mrow><mi>K</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mo>*</mo></mrow></msup></mrow></math> system

We have studied the <math><mrow><msub><mrow><mi>Ω</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>→</mo><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><msup><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>,</mo><mi>η</mi><mo>)</mo><mi>π</mi><msup><mrow><mi>Ξ</mi></mrow><mrow><mo>*</mo></mrow></msup></mrow></math> and <math><mrow><msub><mrow><mi>Ω</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>→</mo><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><msup><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>,</mo><mi>η</mi><mo>)</mo><mover><mrow><mi>K</mi></mrow><mrow><mo>¯</mo></mrow></mover><msup><mrow><mi>Σ</mi></mrow><mrow><mo>*</mo></mrow></msup></mrow></math> decays, where the final <math><mi>π</mi><msup><mi>Ξ</mi><mo>*</mo></msup></math> or <math><mover><mi>K</mi><mo>¯</mo></mover><msup><mi>Σ</mi><mo>*</mo></msup></math> comes from the decay of two resonances around the nominal <math><mi>Ξ</mi><mo>(</mo><mn>1820</mn><mo>)</mo></math>, which are generated from the interaction of coupled channels made of a pseudoscalar and a baryon of the decuplet. The <math><mi>π</mi><msup><mi>Ξ</mi><mo>*</mo></msup></math> mass distributions obtained in the six different reactions studied are quite different, and we single out four of them, which are free of a tree level contribution, showing more clearly the effect of the resonances. The lower mass resonance is clearly seen as a sharp peak, but the higher mass resonance manifests itself through an interference with the lower one that leads to a dip in the mass distribution around 1850 MeV. Such a feature is similar to the dip observed in the <math><mi>S</mi></math>-wave <math><mi>π</mi><mi>π</mi></math> cross section around the 980 MeV coming from the interference of the <math><msub><mi>f</mi><mn>0</mn></msub><mo>(</mo><mn>500</mn><mo>)</mo></math> and <math><msub><mi>f</mi><mn>0</mn></msub><mo>(</mo><mn>980</mn><mo>)</mo></math> resonances. Its observation in coming upgrades of present facilities will shed light on the existence of these two resonances and their nature. On the other hand, when the <math><mrow><msub><mrow><mi>Ω</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>→</mo><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><msup><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>,</mo><mi>η</mi><mo>)</mo><mover><mrow><mi>K</mi></mrow><mrow><mo>¯</mo></mrow></mover><msup><mrow><mi>Σ</mi></mrow><mrow><mo>*</mo></mrow></msup></mrow></math> reactions are studied, both peaks are observed.

<math><mrow><msub><mrow><mi>Ω</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>→</mo><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><msup><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>,</mo><mi>η</mi><mo>)</mo><mi>π</mi><msup><mrow><mi>Ξ</mi></mrow><mrow><mo>*</mo></mrow></msup></mrow></math>, <math><mrow><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><msup><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>,</mo><mi>η</mi><mo>)</mo><mover><mrow><mi>K</mi></mrow><mrow><mo>¯</mo></mrow></mover><msup><mrow><mi>Σ</mi></mrow><mrow><mo>*</mo></mrow></msup></mrow></math> reactions and the two <math><mi>Ξ</mi><mo>(</mo><mn>1820</mn><mo>)</mo></math> states

The <math><msup><mi>N</mi><mo>*</mo></msup><mo>(</mo><mn>1535</mn><mo>)</mo></math> can be dynamically generated in the chiral unitary approach with the coupled channels, <math><mrow><msup><mrow><mi>K</mi></mrow><mrow><mn>0</mn></mrow></msup><msup><mrow><mi>Σ</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>,</mo><msup><mrow><mi>K</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>Σ</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>,</mo><msup><mrow><mi>K</mi></mrow><mrow><mo>+</mo></mrow></msup><mi>Λ</mi></mrow></math>, and <math><mi>η</mi><mi>p</mi></math>. In this work, we evaluate the correlation functions for every channel and face the inverse problem. Assuming the correlation functions to correspond to real measurements, we conduct a fit to the data within a general framework in order to extract the information contained in these correlation functions. The bootstrap method is used to determine the uncertainties of the different observables, and we find that, assuming errors of the same order than in present measurements of correlation functions, one can determine the scattering length and effective range of all channels with a very good accuracy. Most remarkable is the fact that the method predicts the existence of a bound state of isospin <math><mfrac><mn>1</mn><mn>2</mn></mfrac></math> nature around the mass of the <math><msup><mi>N</mi><mo>*</mo></msup><mo>(</mo><mn>1535</mn><mo>)</mo></math> with an accuracy of 6 MeV. These results should encourage the actual measurement of these correlation functions (only the <math><msup><mi>K</mi><mo>+</mo></msup><mi>Λ</mi></math> one is measured so far), which can shed valuable light on the relationship of the <math><msup><mi>N</mi><mo>*</mo></msup><mo>(</mo><mn>1535</mn><mo>)</mo></math> state to these coupled channels, a subject of continuous debate.

Chu-Wen Xiao

Correlation functions for the <math><mrow><msup><mrow><mi>N</mi></mrow><mrow><mo>*</mo></mrow></msup><mo>(</mo><mn>1535</mn><mo>)</mo></mrow></math> and the inverse problem

We study the light and heavy quark mass dependence of the low-lying charmed mesons in the framework of one-loop HHχPT. The low energy constants are determined by analyzing the available lattice data from different LQCD simulations. Model selection tools are implemented to determine the relevant parameters as required by data with a higher precision. Discretization and other effects due to the charm quark mass setting are discussed.

Fernando Gil-Domínguez

Quark mass dependence of the low-lying charmed mesons at one loop in HH Pt

We study the <math><mrow><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>*</mo></mrow></msubsup><msup><mrow><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mo>*</mo></mrow></msup></mrow></math> system in connection with the <math><mi>J</mi><mo>/</mo><mi>ψ</mi><msup><mi>K</mi><mo>*</mo></msup></math> in coupled channels and observe that, within reasonable values of the cutoff used to regularize the loops, the system does not develop a bound state. However, the <math><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><msup><mn>2</mn><mo>+</mo></msup></math> channel has enough attraction to create a strong cusp structure that shows up in the <math><mi>J</mi><mo>/</mo><mi>ψ</mi><msup><mi>K</mi><mo>+</mo></msup></math> invariant mass distribution in the <math><msup><mi>B</mi><mo>+</mo></msup><mo>→</mo><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>ϕ</mi><msup><mi>K</mi><mo>+</mo></msup></math> decay at the <math><msubsup><mi>D</mi><mi>s</mi><mo>*</mo></msubsup><msup><mover><mi>D</mi><mo>¯</mo></mover><mo>*</mo></msup></math> threshold. Such structure is visible in the experimental <math><msup><mi>B</mi><mo>+</mo></msup><mo>→</mo><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>ϕ</mi><msup><mi>K</mi><mo>+</mo></msup></math> and <math><msup><mover><mi>B</mi><mo>¯</mo></mover><mn>0</mn></msup><mo>→</mo><mi>J</mi><mo>/</mo><mi>ψ</mi><msup><mi>K</mi><mo>+</mo></msup><msup><mi>K</mi><mo>−</mo></msup></math> decays, with small statistics, and our results should stimulate further measurements around this region, given the fact that cusp effects provide as valuable information on hadron dynamics as resonances themselves.

Japan

Natsumi Ikeno

<math><msub><mi>Z</mi><mrow><mi>c</mi><mi>s</mi></mrow></msub></math> states from the <math><msubsup><mi>D</mi><mi>s</mi><mo>*</mo></msubsup><msup><mover><mi>D</mi><mo>¯</mo></mover><mo>*</mo></msup></math> and <math><mi>J</mi><mo>/</mo><mi>ψ</mi><msup><mi>K</mi><mo>*</mo></msup></math> coupled channels: Signal in <math><msup><mi>B</mi><mo>+</mo></msup><mo>→</mo><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>ϕ</mi><msup><mi>K</mi><mo>+</mo></msup></math> decay

We study the <math><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>→</mo><msup><mrow><mi>K</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>⁎</mo><mo>−</mo></mrow></msubsup><msup><mrow><mi>D</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>+</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>−</mo></mrow></msubsup><msup><mrow><mi>D</mi></mrow><mrow><mo>⁎</mo><mn>0</mn></mrow></msup><mo>)</mo></math> reaction recently measured at BESIII, from where a new <math><msub><mrow><mi>Z</mi></mrow><mrow><mi>c</mi><mi>s</mi></mrow></msub></math> state has been reported. We study the interaction of <math><msub><mrow><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mi>s</mi></mrow></msub><msup><mrow><mi>D</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math> with the coupled channels <math><mi>J</mi><mo>/</mo><mi>ψ</mi><msup><mrow><mi>K</mi></mrow><mrow><mo>−</mo></mrow></msup></math>, <math><msup><mrow><mi>K</mi></mrow><mrow><mo>⁎</mo><mo>−</mo></mrow></msup><msub><mrow><mi>η</mi></mrow><mrow><mi>c</mi></mrow></msub></math>, <math><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>−</mo></mrow></msubsup><msup><mrow><mi>D</mi></mrow><mrow><mo>⁎</mo><mn>0</mn></mrow></msup></math>, <math><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>⁎</mo><mo>−</mo></mrow></msubsup><msup><mrow><mi>D</mi></mrow><mrow><mn>0</mn></mrow></msup></math> by means of an extension to the charm sector of the local hidden gauge approach. We find that the <math><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>−</mo></mrow></msubsup><msup><mrow><mi>D</mi></mrow><mrow><mo>⁎</mo><mn>0</mn></mrow></msup><mo>+</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>⁎</mo><mo>−</mo></mrow></msubsup><msup><mrow><mi>D</mi></mrow><mrow><mn>0</mn></mrow></msup></math> combination couples to <math><mi>J</mi><mo>/</mo><mi>ψ</mi><msup><mrow><mi>K</mi></mrow><mrow><mo>−</mo></mrow></msup></math> and <math><msup><mrow><mi>K</mi></mrow><mrow><mo>⁎</mo><mo>−</mo></mrow></msup><msub><mrow><mi>η</mi></mrow><mrow><mi>c</mi></mrow></msub></math>, but the <math><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>−</mo></mrow></msubsup><msup><mrow><mi>D</mi></mrow><mrow><mo>⁎</mo><mn>0</mn></mrow></msup><mo>−</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>⁎</mo><mo>−</mo></mrow></msubsup><msup><mrow><mi>D</mi></mrow><mrow><mn>0</mn></mrow></msup></math> combination does not. The coupled channels help to build up strength in the <math><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>−</mo></mrow></msubsup><msup><mrow><mi>D</mi></mrow><mrow><mo>⁎</mo><mn>0</mn></mrow></msup><mo>+</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>⁎</mo><mo>−</mo></mrow></msubsup><msup><mrow><mi>D</mi></mrow><mrow><mn>0</mn></mrow></msup></math> diagonal scattering matrix close to threshold and, although the interaction is not strong enough to produce a bound state or resonance, it is sufficient to produce a large accumulation of strength at the <math><msub><mrow><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mi>s</mi></mrow></msub><msup><mrow><mi>D</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math> threshold in the <math><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>→</mo><msup><mrow><mi>K</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>⁎</mo><mo>−</mo></mrow></msubsup><msup><mrow><mi>D</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>+</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>−</mo></mrow></msubsup><msup><mrow><mi>D</mi></mrow><mrow><mo>⁎</mo><mn>0</mn></mrow></msup><mo>)</mo></math> reaction in agreement with experiment.

The (3985) as a Threshold Effect From the Interaction

We evaluate the contribution to the X(3872) width from a triangle mechanism in which the X decays into $$D^{*0}{\bar{D}}^0 -cc$$ <math><mrow><msup><mi>D</mi><mrow><mrow></mrow><mo>∗</mo><mn>0</mn></mrow></msup><msup><mrow><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mn>0</mn></msup><mo>-</mo><mi>c</mi><mi>c</mi></mrow></math> , then the $$D^{*0} ({\bar{D}}^{*0})$$ <math><mrow><msup><mi>D</mi><mrow><mrow></mrow><mo>∗</mo><mn>0</mn></mrow></msup><mrow><mo>(</mo><msup><mrow><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mrow></mrow><mo>∗</mo><mn>0</mn></mrow></msup><mo>)</mo></mrow></mrow></math>  decays into $$D^0 \pi ^0$$ <math><mrow><msup><mi>D</mi><mn>0</mn></msup><msup><mi>π</mi><mn>0</mn></msup></mrow></math>  ($${\bar{D}}^0 \pi ^0$$ <math><mrow><msup><mrow><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mn>0</mn></msup><msup><mi>π</mi><mn>0</mn></msup></mrow></math> ) and the $$D^0 {\bar{D}}^0$$ <math><mrow><msup><mi>D</mi><mn>0</mn></msup><msup><mrow><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mn>0</mn></msup></mrow></math>  fuse to produce $$\pi ^+ \pi ^-$$ <math><mrow><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>-</mo></msup></mrow></math> . This mechanism produces an asymmetric peak from a triangle singularity in the $$\pi ^+ \pi ^-$$ <math><mrow><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>-</mo></msup></mrow></math>  invariant mass with a shape very sensitive to the X mass. We evaluate the branching ratios for a reaction where this effect can be seen in the $$B^- \rightarrow K^- \pi ^0 \pi ^+ \pi ^-$$ <math><mrow><msup><mi>B</mi><mo>-</mo></msup><mo>→</mo><msup><mi>K</mi><mo>-</mo></msup><msup><mi>π</mi><mn>0</mn></msup><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>-</mo></msup></mrow></math>  reaction and show that the determination of the peak in the invariant mass distribution of $$\pi ^+ \pi ^-$$ <math><mrow><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>-</mo></msup></mrow></math>  is all that is needed to determine the X mass. Given the present uncertainties in the X mass, which do not allow to know whether the $$D^{*0} {\bar{D}}^0$$ <math><mrow><msup><mi>D</mi><mrow><mrow></mrow><mo>∗</mo><mn>0</mn></mrow></msup><msup><mrow><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mn>0</mn></msup></mrow></math>  state is bound or not, measurements like the one suggested here should be most welcome to clarify this issue.

Triangle singularity in $$B^-\rightarrow K^-X(3872);X\rightarrow \pi ^0\pi ^+\pi ^-$$ <math><mrow><msup><mi>B</mi><mo>-</mo></msup><mo>→</mo><msup><mi>K</mi><mo>-</mo></msup><mi>X</mi><mrow><mo>(</mo><mn>3872</mn><mo>)</mo></mrow><mo>;</mo><mi>X</mi><mo>→</mo><msup><mi>π</mi><mn>0</mn></msup><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>-</mo></msup></mrow></math>  and the X(3872) mass

In a recent paper [1], the BESIII Collaboration reported the so-called first observation of pure W-annihilation decays <math><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>→</mo><msubsup><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>+</mo></mrow></msubsup><mo>(</mo><mn>980</mn><mo>)</mo><msup><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msup></math> and <math><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>→</mo><msubsup><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>0</mn></mrow></msubsup><mo>(</mo><mn>980</mn><mo>)</mo><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup></math>. The measured absolute branching fractions are, however, puzzlingly larger than those of other measured pure W-annihilation decays by at least one order of magnitude. In addition, the relative phase between the two decay modes is found to be about 180 degrees. In this letter, we show that all these can be easily understood if the <math><msub><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mn>980</mn><mo>)</mo></math> is a dynamically generated state from <math><mover><mrow><mi>K</mi></mrow><mrow><mo>¯</mo></mrow></mover><mi>K</mi></math> and πη interactions in coupled channels. In such a scenario, the <math><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math> decay proceeds via internal W emission instead of W-annihilation, which has a larger decay rate than W-annihilation. The proposed decay mechanism and the molecular nature of the <math><msub><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mn>980</mn><mo>)</mo></math> also provide a natural explanation to the measured negative interference between the two decay modes.